Can you really measure distance between two points?

March 2, 2019  393 words 2 mins read  Comments

Of course! By using a ruler, right?

But in reality, you are actually coinciding given two points with the points of your ruler. Then you will say, these points are separated by a $x$ unit in length.

I want to mention now, in the above line consisting of points [see above figure], these points are equally separated. In order to make an equal separation, you do not need a standard ruler (i.e. with standard unit). Is that right? This means the line is a ruler but without unit.

Now, you can count all the points from a coinciding point with A to coinciding point with B including points lies between A and B in our new ruler then, you will say the distance between A and B is 6 (because in the ruler, counting starts from 0). But, where is its unit? Does 6 really mean the distance? No! This 6 really means the line coincided 7 points between A and B including themselves. Otherwise, nothing. We can even make more small spacing with more points in the line. So, how can we track different spacing and relationship between them? The solution is to assign “unit” to each and every spacing. You cannot simply measure spacing with itself. But, you can measure the ratio of the spacing to a standard ruler such that it gives one because of the coinciding concept.

Generally, if one measures a length, one actually measures the ratio of that length to a standard unit of length. i.e.

$ \frac{\text{Distance between AB}}{6 cm \text{ in centimeter ruler}} = 1$

$\therefore \text{Distance between AB} = 6 cm$.

This is how AB gets its unit.

In Quantum Gravity, we are interested to probe the spacetime even below the Planck length (order of $10^{-35}$m) but, we don’t know how to define spacing below that length. In reality, we don’t know does spacetime at that region is discrete or continuum? But, most of the candidate theory for Quantum Gravity believes that spacetime has a discrete nature, and even length and time have discreteness. So, if the length is discrete how to define a spacing between two points can be a major problem in Quantum Gravity. Because we define spacing AB supposing that it can include infinitely many points between A and B.

Lastly, this post is dedicated to Dr. Joshua Cooperman’s article arXiv:1604.01798.

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  • Damodar Rajbhandari
    Written by Damodar Rajbhandari, a PhD student in Theoretical Physics at the Jagiellonian University, Poland, working in the Causal Dynamical Triangulations group at the Institute of Theoretical Physics.
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